Positive Semi-Definite and Sum of Squares Biquadratic Polynomials - Chunfeng Cui; Liqun Qi; Yi Xu (Mathematics)
Hilbert proved in 1888 that a positive semi-definite (PSD) homogeneous quartic polynomial of three variables always can be expressed as the sum of squares (SOS) of three quadratic polynomials, and a psd homogeneous quartic polynomial of four variables may not be sos. Only after 87 years, in 1975, Choi gave the explicit expression of such a psd-not-sos (PNS) homogeneous quartic polynomial of four variables. An <semantics>m×n</semantics> biquadratic polynomial is a homogeneous quartic polynomial of <semantics>m+n</semantics> variables. In this paper, we show that an <semantics>m×n</semantics> biquadratic polynomial can be expressed as a tripartite homogeneous quartic polynomial of <semantics>m+n−1</semantics> variables. Therefore, by Hilbert’s theorem, a <semantics>2×2</semantics> PSD biquadratic polynomial can be expressed as the sum of squares of three quadratic polynomials. This improves the result of Calderón in 1973, who proved that a <semantics>2×2</semantics> biquadratic polynomial can be expressed as the sum of squares of nine quadratic polynomials. Furthermore, we present a necessary and sufficient condition for an <semantics>m×n</semantics> psd biquadratic polynomial to be sos, and show that if such a polynomial is sos, then its sos rank is at most <semantics>mn</semantics>. Then we give a constructive proof of the sos form of a <semantics>2×2</semantics> psd biquadratic polynomial in three cases.